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IV

.. , , 19.04.2012



1. . 2. . ( ). 3. . 4. : лЁ . 5. н .
is



( ) kram

k =

eff

exp{-Ea / k BT }




(, )

H. A. Kramers ( )

. . ( )

... effects








- - -

definit


:


smol

Pi

(f)

( q, )

- () .

(D):

k BT D= m
k BT D= 2s

-

L

-


:

dP(q,) 1 dU(q) i =D + P(q,) i qq kBT dq d
: H.A. Kramers, Physica, 7 (1940) 284-304.

1 L >> veff

dP(q,) i =0 d Pi (q ) 1 dU (q ) + J (q) = D Pi (q ) q k BT dq
k =- J (q )dq = J (q)dq
0 1 1 0





cont


U (q) J (q ) = D exp - Pi (q ) ex k BT q
q U (q Pi (q ) exp k BT U (q) exp k BT

U (q) p k BT



J (q) = D

)



s q 2 Pi (q ) = exp - k BT k BT

s

cont


Ea k BT k 2D exp - k BT s L k BT

s

Ea exp - k BT k BT

s

k

1



k BT

L

(s +I ) 2 exp - s 4s k BT

table


sugar


?



лЁ :

C D ( w) = + + 1 + (iw C ) 1 + iw

D
EG


: - (EG)

-EG:

3 1 2 + + + ( ) = 1 + i 2 1 1 + i 2 2 1 + i 2 3
"- "



l_in


(q1 ) ( q2 ); .

Uf (q1, q2 ) = s (q1 -1)2 +in (q2 -1)2 + I Ui (q1, q2 ) = q + q ;
2 s1 2 in 2

P(q1, q2 ; )

= div j

j =(j

solv

, jin )

j

solv

P (q1 , q2 ; ) U (q1 , q2 ) 1 = D1 + P (q1 , q2 ; ) q1 q1 k BT

P (q1 , q2 ; ) U (q1 , q2 ) 1 + jin = D2 P (q1 , q2 ; ) q2 q2 k BT

in fast


! q - () q1- ()
P Pi(q,) Pf(q,) н (i) (f) . E(q, q1)
ovch

:


. . , . . , 17 (1982) 507.

d 1 d Pq,) =D + U(q)P(q,)-kif (qPq,)+kfi (qP (q,) ) i( )f i( i d qq kBT dq

d 1 d P(,)=D + U(q)P(q,)-kfi(q)P(,)+kif (q)P(q,) fq f fq i d qq kBT dq
U(q)=E(q, D н kif(q) kfi(q) нлЁ ( 0); ; лЁ ).
sumi


H. Sumi and R. Marcus
J. Chem. Phys. 84 (1985) 4894.

: k fi << 1

d 1 d P(q,) = D + U(q)P(q,) -kif (q)P(q,) i i i d q q kBT dq
лЁ

k BT D= 2s

k in = in exp - E * (q ) / kT
L

{

}

E * (q ) = E(q, q

* saddle

(q)) - E(q,0)

vin 10 c
13

-1

rate


(k) ?
!

k = 1/

qR



0 qL

P (q, )d dq = 1/ P * (q)dq i
qL


qR

P * (q ) = P (q, )d
0

P*(q) (.) лЁ,

: (H.Sumi,R.Markus)
q

kif(q)P*(q)

( )

k=

q q



R

P *(q )dq

L

q



R

P (q, )dq
avoidance

L



(q, qin), -.

s = 0.5 eV , in =1.2 eV

edta


(in /), [Cr(EDTA)]- and [Co(EDTA)]Redox pair

in
0.66 1.79

in
0.6 1.93

in
0.63 1.86

[Cr(EDTA)]

-/2-

[Co(EDTA)]

-/2-

[Co(EDTA)]- [Cr(EDTA)]Co(edta)


lgk ( s )

[Cr(EDTA)]lgk ( s )
10.8
-1

[Co(EDTA)]
-1

-

6.0

10.4

5.5

10.0

5.0

9.6 -12.75 -12.00

4.5

lgs (s)

-11.25

-10.50

-12.75

-12.00

lgs (s)

-11.25

-10.50

log ks vs. logs,, [Cr(EDTA)]- [Co(EDTA)]s = 0.3 ; =0.1 V (solid); =0.2 V (dashed); =0.3 V (dotted). Co(EDTA)-
brown


()

dz 1 U ( z ) dz 1 =- - + Frand ( ) 2 d m x d m
2
лЁ

< Frand (0), Frand ( ) >= A ( )
P. Langevin

A = k BT m

-

real


:
Allen M.P., Tildesley D.J. Computer Simulation of Liquids, Clarendon Press, Oxford, 1987

z ( + ) = z ( ) + c1 v( ) + c2 2 a( ) + z


rand

v( + ) = c0 v( ) + c1 a( ) + vrand
c0 = exp(- )





c1 = ( ) -1 (1 - c0 )

c2 = ( )-1 (1 - c1 )

z

rand

vrand

- ( );

distrib



z, k



12 10 8 6 4

z,

2 0 -2 -4

1 = N


i =1

N



-6

,
0 1000 2000 3000 4000 5000

i

t,



k=

1





k = kTST


kast


лЁ
S.M. Kast et al. J. Chem. Phys. 100 (1994) 566-576

m1 н ; m2 н ;

m2 = m1 + m2

=

2





v н ; u н ;

vn +1 - vn = 2 (un - vn )
m2 f (u ) = 2 k BT
1/ 2

m2u 2 exp - 2 k BT
real




:

rn +1 = rn + vn + an /2

vn = vn -1 + vn + (an + an +1 ) / 2
an = 2



(un - vn -1 )
( )

Treal = Tvirt



1) , . .

2)
н . (. . ). 3) н . 4) , .